Dear walker,

In page 89 there are some mistakes.

SLOPE AND QUADRANTS:
1. If the slope of a line is negative, the line WILL intersect quadrants II and IV. X and Y intersects of the line with negative slope have the same sign. Therefore if X and Y intersects are positive, the line intersects quadrant I; if negative, quadrant III.
2. If the slope of line is positive, line WILL intersect quadrants I and III. Y and X intersects of the line with positive slope have opposite signs. Therefore if X intersect is negative, line intersects the quadrant II too, if positive quadrant IV.
3. Every line (but the one crosses origin (0,0) OR parallel to X or Y axis OR X and Y axis themselves) crosses three quadrants. Only the line which crosses origin OR is parallel to either of axis crosses only two quadrants.
4. If a line is horizontal it has a slope of 0, is parallel to X-axis and crosses quadrant I and II if the Y intersect is positive OR quadrants III and IV, if the Y intersect is negative. Equation of such line is y=b, where b is y intersect.
5. If a line is vertical, the slope is not defined, line is parallel to Y-axis and crosses quadrant I and IV, if the X intersect is positive and quadrant II and III, if the X intersect is negative. Equation of such line is x=a, where a is x-intercept.

In addition, I would like to suggest some changes to the explanation of Lines in Coordinate Geometry (page 85). I think there are too many formulas and a systematic approach is not encouraged. For instance:

Page 85

I think using a b c on General Form is confusing. I would recommend to use A B C, such as:

Ax+By+C=0

The point intercept form is given right away, together with the slope and the y-intercept for the General Form, all using the same letters. I would change the whole explanation to something like:

From the General Form Ax+By+C=0,

By=-Ax-C

\frac{B}{B}y=\frac{-A}{B}x+\frac{-C}{B}

y=\frac{-A}{B}x+\frac{-C}{B}

Where \frac{-A}{B} is the slope and \frac{-C}{B} is the y-intercept. From here:

y=mx+b.

This is a much more intuitive approach and does not require to remember 3 formulas.

Page 86

Why give the formula of a line passing through two points? Following an intuitive approach, I would just:

1. Calculate slope from the line crossing the two given points.
2. Plug in the slope and one of the points in the general equation to get b. No formulas required.

Page 87

Again, another formula is given. Same approach as above can be used with (x,0) and (0,y).

Page 92

Another formula is given. Why not just use the same slope and the point?

Maybe all the formulas can be included at the end as an extra, but using most of them right away somewhat breaks the logic and just give a takeaway that does not ensure that the concept is correctly understood. I think it's much more powerful (and simple!) to use logic and know where everything is coming from.

That come to mind, the basic and most important are: Distance, Mid Point, General Form, Point-Intercept Form, Slope of a Line, Parallel lines = same slope, Perpendicular lines = negative reciprocal, Distance Line-Point.


Best Regards, and great job on this Math Book!

jorisboris, I am not sure if I understand you, but so far I think you are misunderstanding the formulas (are you referring to the formulas labeled as 1), 2), 3) and 4) in p103?)

Translated to words:

1) If the number you add to the set (y) is higher than the standard deviation (in formulas: if the distance from y to the median is higher than the distance of the standard deviation to the median, if y > STD) then the standard deviation of the set will increase (in formulas: the new standard deviation will be higher than the old standard deviation).

Or what is the same: new numbers added to the set will reduce the STD if they present less deviation than the STD (they are closer to the mean than the STD range), will increase the STD if they are further away from the mean than the range of the STD, will keep the STD the same if they are exactly as far away from the mean as the STD and will minimize as much as possible the STD if they are exactly equal to the mean (they add a deviation of 0, so the STD has the same total deviation, +0 is added) but more term to divide, thus decreases as much possible). Of course if you have 5 terms and you add a 6th term to the set that is exactly the same as the mean the STD will decrease much more than if you have 1000 terms with a huge STD and you just add one new element to the set. Still, the behavior will be the same, as these considerations are QUALITATIVE, not QUANTITATIVE.

Check my notes attached. In summary, these formulas only explain the behavior of the Standard Deviation based on the kind of number that you add to the set. Colors match, so line 1) refers to red point, line 2) refers to orange point, etc.

Does this solve your question?

Page 58, bottom:
Where b is the length of the base, a and c the other sides; h is the length of the corresponding altitude; R is
the Radius of circumscribed circle; r is the radius of inscribed circle; P is the perimeter

The Perimeter of the triangle (a+b+c) or of the in-/circum-scribed circle (2*pi*r /*R)?
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